Sec 4 E-Math Vectors in Two Dimensions topical questions

Write AB as a column vector, where A = (2, 1) and B = (6, 4).

Find the magnitude of the vector (6, 3), correct to 3 significant figures.

Point P(3, 5) undergoes a translation by (−2, 4). Find the image P′.

Given u = (2, −3) and v = (−1, 5), find (a) u + v and (b) u − v.

1. (a)u + v.[1]
2. (b)u − v.[2]

Given a = (2, 1) and b = (4, −2), find 3a − 2b.

Given a = (3, −4), find (a) |a| and (b) |2a|.

Under a translation, the point X(1, 1) maps to X′(4, −3). (a) Write the translation as a column vector. (b) Find the image of Y(−2, 5) under the same translation.

A = (−2, 3) and AB = (5, −1). (a) Find B. (b) Find |AB|, correct to 3 significant figures.

1. (a)Find B.[2]
2. (b)Find |AB|.[2]

ABCD is a parallelogram with A(1, 1), B(4, 2) and D(2, 5). (a) Write AB as a column vector. (b) Using DC = AB, find the coordinates of C.

1. (a)AB.[2]
2. (b)C.[2]

O is the origin, A = (2, 0) and B = (0, 3). The point C satisfies OC = OA + OB. (a) Write down the coordinates of C. (b) Describe the quadrilateral OACB.

1. (a)C.[2]
2. (b)Quadrilateral OACB.[2]

M is the midpoint of AB, where a = (2, 3) and b = (6, 7). (a) Find OM. (b) Find |OM|, correct to 3 significant figures.

1. (a)OM.[2]
2. (b)|OM|.[2]

Given a = (2, 1), b = (1, 3) and ma + nb = (5, 5), find the values of m and n.

O is the origin, OA = (3, 1) and OB = (1, 4). The point C satisfies OC = OA + OB. (a) Find C. (b) Using vectors, show that OACB is a parallelogram.

1. (a)Find C.[2]
2. (b)Parallelogram proof.[2]

Given a = (4, −2) and b = (−2, 6): (a) find the vector 2a + b and its magnitude, correct to 3 significant figures; (b) show that 2a + b is parallel to (3, 1).

In triangle OPQ, OP = p = (4, 2) and OQ = q = (2, 8). M is the midpoint of PQ. (a) Express OM in terms of p and q. (b) Find OM as a column vector. (c) Find |OM|, correct to 3 significant figures.

1. (a)OM in terms of p and q.[2]
2. (b)OM as a column vector.[2]
3. (c)|OM|.[1]

A(1, 1), B(3, 2), C(6, 1) and D(4, 0) are four points. (a) Write AB and DC as column vectors. (b) Explain what your answers show about ABCD. (c) Write AD as a column vector and state whether AB = AD.

1. (a)AB and DC.[2]
2. (b)Conclusion about ABCD.[2]
3. (c)AD and comparison.[2]

P(1, 2), Q(4, 6) and R(7, 10) are three points. (a) Write PQ and QR as column vectors. (b) Show that P, Q and R are collinear. (c) State the relationship between Q and the segment PR.

1. (a)PQ and QR.[2]
2. (b)Collinearity.[2]
3. (c)Q on PR.[2]

OABC is a parallelogram with OA = a = (3, 1) and OC = c = (1, 4). M is the midpoint of the diagonal AC. (a) Write down OB in terms of a and c. (b) Write down OM in terms of a and c, and explain the relationship between OB and OM. (c) Find OM as a column vector and its magnitude, correct to 3 significant figures.

1. (a)OB.[2]
2. (b)OM and its link to OB.[2]
3. (c)OM and |OM|.[2]

OPQR is a parallelogram with O the origin, P(2, 1) and R(5, 4), where OQ is a diagonal. (a) Find the coordinates of Q. (b) M is the midpoint of diagonal OQ and N is the midpoint of diagonal PR. Show that M and N coincide, and state what this demonstrates.

1. (a)Find Q.[2]
2. (b)Midpoints and conclusion.[4]

A(1, 2), B(3, 5) and C(7, 11) are three points. (a) Express AB, BC and AC as column vectors. (b) Prove that A, B and C are collinear. (c) State the ratio AB : BC and the position of B on the line segment AC.

1. (a)Column vectors.[2]
2. (b)Collinearity proof.[2]
3. (c)Ratio and position.[2]